Incremental equations in curvature-dependent surface elasticity

TL;DR

This paper introduces a comprehensive incremental framework for hyperelastic solids with surfaces that depend on stretch and curvature, enabling analysis of stability and instabilities in systems where surface curvature influences mechanical behavior.

Contribution

It develops a variational, coordinate-free formulation for curvature-dependent surface elasticity and extends classical small-on-large theory to include surface-curvature-induced stresses.

Findings

Derived governing equations valid for arbitrary geometries.

Analyzed instability thresholds in coated cylindrical substrates.

Demonstrated influence of surface effects on stability in elasto-capillary systems.

Abstract

We develop a general incremental framework for hyperelastic solids whose surfaces exhibit both stretch-dependent and curvature-dependent elastic behavior. Building upon a variational formulation of curvature-dependent surface elasticity, we derive compact governing equations expressed in a coordinate-free Lagrangian setting that remain valid for arbitrary geometries. Linearization about an arbitrarily large finite deformation yields incremental bulk and surface balance laws that closely resemble the classical small-on-large theory, but are now extended to include surface-curvatureinduced stresses. The applicability of the general theory is demonstrated by analyzing the onset of periodic beading in a soft cylindrical substrate coated with a surface layer exhibiting stretching- or curvature-dependent behavior, illustrating how surface stretching and bending effects influence instability thresholds for both compressible and incompressible bulk. This unified formulation thus provides a foundation for studying stability phenomena in elasto-capillary systems where surface curvature plays a critical mechanical role.